Preheat oven to 350 degrees. Line a baking pan with parchment paper and set aside.
Prepare the mushrooms by cleaning the shiitake caps with a slightly damp paper towel. Discard stems. Combine the 1 tablespoon olive oil, Worcestershire sauce, salt and pepper in a bowl. Add the mushroom caps and toss gently to combine. Place the mushrooms, stem side down, on the prepared sheet pan and roast for 8 minutes. Remove pan from oven and flip mushrooms over.
While mushrooms are roasting prepare the stuffing. Melt the 1 tablespoon butter in a large sauté pan. Add the shallot and garlic and sauté for one minute. Add the spinach, the white wine and the lemon juice and cook for 1-to-2 minutes to wilt, tossing several times. Remove from the heat and let cool to the touch. Strain off any extra liquid.
Combine the panko bread crumbs, 2 tablespoons shredded Parmesan or Asiago cheese, the remaining 2 tablespoons melted butter and the chopped Italian parsley leaves.
Divide the spinach mixture among the 12 mushroom caps. Top each with some of the panko mixture. Drizzle the tops with the remaining 1 tablespoon melted butter. Flash the mushrooms under a broiler turned on high, watching carefully. The tops of the bread crumbs should turn a toasty brown color. Remove and place on a serving platter. Top with the remaining 2 tablespoons of shredded Parmesan or Asiago cheese.